Problem: $\dfrac{ 4l - 3m }{ 8 } = \dfrac{ -7l - 6n }{ -6 }$ Solve for $l$.
Solution: Multiply both sides by the left denominator. $\dfrac{ 4l - 3m }{ {8} } = \dfrac{ -7l - 6n }{ -6 }$ ${8} \cdot \dfrac{ 4l - 3m }{ {8} } = {8} \cdot \dfrac{ -7l - 6n }{ -6 }$ $4l - 3m = {8} \cdot \dfrac { -7l - 6n }{ -6 }$ Multiply both sides by the right denominator. $4l - 3m = 8 \cdot \dfrac{ -7l - 6n }{ -{6} }$ $-{6} \cdot \left( 4l - 3m \right) = -{6} \cdot 8 \cdot \dfrac{ -7l - 6n }{ -{6} }$ $-{6} \cdot \left( 4l - 3m \right) = 8 \cdot \left( -7l - 6n \right)$ Distribute both sides $-{6} \cdot \left( 4l - 3m \right) = {8} \cdot \left( -7l - 6n \right)$ $-{24}l + {18}m = -{56}l - {48}n$ Combine $l$ terms on the left. $-{24l} + 18m = -{56l} - 48n$ ${32l} + 18m = -48n$ Move the $m$ term to the right. $32l + {18m} = -48n$ $32l = -48n - {18m}$ Isolate $l$ by dividing both sides by its coefficient. ${32}l = -48n - 18m$ $l = \dfrac{ -48n - 18m }{ {32} }$ All of these terms are divisible by $2$ $l = \dfrac{ -{24}n - {9}m }{ {16} }$